On questions 1 and 2: how would you handle reflexive coequalizers with the graphs-as-shapes approach? I agree that it can be done, but isn't it more natural with categories as shapes?

On question 3: there's a point in Mac Lane and Moerdijk's Sheaves in Geometry and Logic where they implicitly use the graphs approach. Lemma VII.6.1 states that a small category is filtering if and only if every finite diagram on it admits a cone. They prove "only if" by induction on the number of nonidentity arrows in the diagram. When I read it, I thought it was a mistake, for obvious reasons: you can't do induction on categories like that. But when I put this to Moerdijk, he said that they were really using the graphs-as-shapes viewpoint, which would make induction possible. I'm not convinced that what they've done is entirely respectable, but it's an answer to your question.

Question 4: in my Basic Category Theory, I used the categories-as-shapes approach. Honestly, I don't think I gave it a lot of thought: I just went with what I'd been brought up with. But if I imagine rewriting the book with graphs as shapes, my first objection would be that it introduces a new concept: graph. At present, the word "graph" doesn't appear once in the whole book. Unnecessarily introducing an extra concept seems like a step backwards. (Which is not to say there couldn't be advantages outweighing this disadvantage.)